A lattice Boltzmann method for axisymmetric

multicomponent flows with high viscosity ratio

Haihu Liua,∗, Lei Wub, Yan Baa, Guang Xia, Yonghao Zhangb

aSchool of Energy and Power Engineering, Xi’an Jiaotong University, 28 West Xianning

Road, Xi’an 710049, ChinabJames Weir Fluids Laboratory, Department of Mechanical & Aerospace Engineering,

University of Strathclyde, Glasgow G1 1XJ, UK

Abstract

A color-gradient lattice Boltzmann method (LBM) is proposed to simulate ax-

isymmetric multicomponent flows. This method uses a collision operator that

is a combination of three separate parts, namely single-component collision op-

erator, perturbation operator, and recoloring operator. A source term is added

into the single-component collision operator such that in each single-component

region the axisymmetric continuity and momentum equations can be exactly re-

covered. The interfacial tension effect is realized by the perturbation operator,

in which an interfacial force of axisymmetric form is derived using the concept

of continuum surface force. A recoloring operator proposed by Latva-Kokko

and Rothman is extended to the axisymmetric case for phase segregation and

maintenance of the interface. To enhance the method’s numerical stability for

handling binary fluids with high viscosity ratio, a multiple-relaxation-time mod-

el is used for the collision operator. Several numerical examples, including static

droplet test, oscillation of a viscous droplet, and breakup of a liquid thread, are

presented to test the capability and accuracy of the proposed color-gradient LB-

M. It is found that the present method is able to accurately capture the phase

interface and produce low spurious velocities. Also, the LBM results are all

in good agreement with the analytical solutions and/or available experimental

∗Corresponding authorEmail address: haihu.liu@xjtu.edu.cn (Haihu Liu)

Preprint submitted to Journal of Computational Physics October 8, 2016

data for a very broad range of viscosity ratios.

Keywords: Lattice Boltzmann method, Axisymmetric flow, Color-gradient

model, High viscosity ratio, Rayleigh instability

1. Introduction

Multiphase, multicomponent flows of incompressible fluids are ubiquitous

in nature and in many industrial processes, and have received considerable at-

tention in the past few decades. With the development of computer hardware

and advances in numerical techniques and algorithms, computational fluid dy-

namics (CFD) has proved to be a parctical and reliable tool for studying and

gaining in-depth insights into the complex behavior of multiphase multicom-

ponent flows. Traditional CFD methods simulate multiphase/multicomponent

flows by solving the macroscopic Navier-Stokes equations (NSEs) together with

a proper technique to track or capture the interface among different fluids.

Generally, these methods are divided into two categories: one is the interface-

tracking method, which uses the Lagrangian approach to explicitly represent the

interface, such as the front-tracking method [1]; and the other is the interface-

capturing method, which uses an indicator function to implicitly represent the

interface in an Eulerian grid, such as the volume-of-fluid (VOF) method [2] and

level set (LS) method [3, 4]. However, the front-tracking method is not suitable

for handling interface breakup and coalescence, because the interface must be

manually ruptured based upon some ad-hoc criteria. The VOF and LS meth-

ods require interface reconstruction or reinitialization to represent/correct the

interface, which may be complex or unphysical. Physically, the interface and

its dynamical behavior are the natural consequence of microscopic interactions

among fluid molecules. Thus, mesoscopic level methods may be better suited to

simulate complex interfacial dynamics in a multiphase/multicomponent system.

The lattice Boltzmann method (LBM), as a mesoscopic method, has been

developed into an alternative to traditional CFD methods for simulating com-

plex fluid flow problems [5, 6]. It is a pseudo-molecular method based on particle

2

distribution functions that performs microscopic operations with mesoscopic ki-

netic equations and reproduces macroscopic behavior [7]. The LBM has several

advantages over traditional CFD methods such as the ability to be programmed

on parallel computers and the ease in dealing with complex boundaries [8]. Be-

sides, its kinetic nature provides many of the advantages of molecular dynamics,

making the LBM particularly useful for simulating multiphase, multicomponent

flows. A number of multiphase, multicomponent models have been proposed in

the LBM community based on the Cartesian coordinate system, and they can

be classified into four major types: color-gradient model [9, 10, 11, 12], phase-

field-based model [13, 14, 15, 16], interparticle-potential model [17, 18, 19], and

mean-field theory model [20]. Among these models, the interparticle-potential

model has recently shown significant improvements in computational stability

and accuracy, see, e.g., Refs. [21, 22, 23, 24, 25]. For a comprehensive review

of these multiphase multicomponent models, interested readers may refer to

Refs. [8, 26, 27].

Among three-dimensional (3D) multiphase multicomponent flows, axisym-

metric flows are special cases that occur when both flow geometry and initial

conditions display axial symmetry. Examples of axisymmetric multiphase multi-

component flows include head on collision of two droplets [28], droplet impact on

a solid surface [29], droplet formation in a co-flow microfluidic device [30], and

so on. To simulate axisymmetric multiphase, multicomponent flows, a direct

way is to apply a 3D multiphase/multicomponent LBM with suitable curved

boundary conditions. Such a treatment, however, does not take the advantage

of the axisymmetric property of the flow and usually needs large computation-

al costs. Alternatively, several researchers have attempted to develop more

effective quasi-two-dimensional (2D) LBMs for simulating axisymmetric mul-

tiphase flows. Premanath and Abraham [31] proposed the first axisymmetric

multiphase LBM based on the mean-field theory model of He et al. [20]. In

their method, some source terms containing density and velocity gradients were

added into the microscopic evolution equations to account for axisymmetric con-

tributions of mass, momentum, and capillary force. By introducing the source

3

terms in the same fashion as Premanath and Abraham [31], Mukherjee and

Abraham [32] extended the high-density-ratio model of Lee and Lin [33] to the

axisymmetric case. Later, Huang et al. [34] presented a phase-field-based hy-

brid method for axisymmetric multiphase flows, in which they used the finite

difference method for solving the axisymmetric Cahn-Hilliard equation and the

multiple-relaxation-time (MRT) LBM for the NSEs. Recently, the interparticle-

potential model of Shan and Chen [17] was extended by Srivastava et al. [35] to

study generic axisymmetric, density-varying flows and multiphase flows. In the

axisymmetric model, a source term containing velocity gradients was introduced

into the lattice Boltzmann equation so that it could match the axisymmetric

NSEs in the Chapman-Enskog analysis, and an axisymmetic Shan-Chen force

was derived by means of a Taylor expansion of its 3D counterpart as well as a

coordinate transformation. More recently, Liang et al. [36] extended their pre-

viously developed phase-field-based model [37] for the solution of axisymmet-

ric multiphase flows. Unlike other axisymmetric multiphase LBMs, the added

source terms that arise from the axisymmetric effect contain no gradients in

this method, thus simplifying the computation and enhancing the computing

efficiency.

As reviewed above, all types of the (Cartesian) multiphase, multicompo-

nent LBMs have been extended to the axisymmetric versions except the color-

gradient model. Compared to other multiphase multicomponent LBMs, the

color-gradient model has its own advantages such as low spurious velocities,

high numerical accuracy, strict mass conservation for each fluid, and good nu-

merical stability for a broad range of fluid properties [38]. In addition, the

color-gradient model has been widely employed to simulate immiscible multi-

component flow problems, in particular those in porous media and microfluidic

devices [39, 40, 41, 42]. Recently, it was also extended to model the thermocap-

illary flows [43, 44] and the contact-line dynamics with the contact angle hys-

teresis [45, 38]. In view of the advantages and great success of the color-gradient

model, it is necessary to develop an axisymmetric version of the color-gradient

LBM that allows for the solution of multicomponent flows at the computational

4

cost of a 2D simulation.

In this work, an axisymmetric multicomponent LBM based on the color-

gradient model is developed. In this method, a source term is added into the

single-component collision operator so that the axisymmetric NSEs could be

correctly reproduced in each single-component region. An interfacial force of

axisymmetric form is derived using the concept of continuum surface force (CS-

F) [46] together with a coordinate transformation, and it is then incorporated

into the LBM using a body force model. A recoloring algorithm proposed by

Latva-Kokko and Rothman [47] is extended to axisymmetric case for produc-

ing phase segregation. In addition, the axisymmetric multicomponent LBM

is implemented in a MRT framework in order to minimize spurious velocities

and increase the numerical stability of solving large viscosity ratio problem-

s [48, 49, 50, 38]. The capability and accuracy of this method are tested by

several typical flow cases, including simulations of a static droplet, oscillation

of a viscous droplet, and breakup of a liquid thread.

2. Theory and Mathematical Model

In this section, an axisymmetric version of the color-gradient multicompo-

nent LBM is presented, and it is developed on the basis of the model of Halliday

and his coworkers [10, 51, 52], which is defined in a Cartesian coordinate system.

There are two fluids, red and blue, considered in the color-gradient LBM. Let

fki (~x, t) represents the particle distribution function (PDF) of the fluid k in the

i-th velocity direction at the position ~x and time t, where k = R or B denotes

the red or blue fluid. The total distribution function is defined as fi = fRi +fB

i .

The time evolution of each colored PDF is a combination of free streaming and

collision:

fki (~x+ ~eiδt, t+ δt) = fk

i (~x, t) + Ωki (~x, t), (1)

where ~ei is the lattice velocity in the ith direction, δt is the time step, and

Ωki is the collision operator. The collision operator consists of three separate

5

parts [11, 45]:

Ωki = (Ωk

i )(3)[

(Ωki )

(1) + (Ωki )

(2)]

, (2)

where (Ωki )

(1) is the single-component collision operator, (Ωki )

(2) is the perturba-

tion operator which contributes to the mixed interfacial region and generates an

interfacial tension, and (Ωki )

(3) represents the recoloring operator which mimics

the phase segregation and keeps the interface sharp.

The single-component collision operator (Ωki )

(1) is designed to recover the

correct macroscopic equations of incompressible axisymmetric flows in each

single-component region. For the axisymmetric flows with an axis in the z-

direction, the continuity and momentum equations in the cylindrical coordi-

nates, in absence of external forces are given by [35, 36]

∂αuα = −ur/r, (3)

and

ρ (∂tuα + uβ∂βuα) = −∂αp+∂β [µ (∂βuα + ∂αuβ)]+µ (∂ruα + ∂αur)

r− 2µur

r2δαr,

(4)

respectively, where α, β indicate the r or z component, and r is the coordinate

in radial direction; uα is the component of velocity in the α direction; p is the

pressure; ρ and µ are the density and dynamic viscosity of the fluid mixture (or

color-blind fluid); and δαβ is the Kronecker delta with two indices. In Eqs.(3)

and (4) we assume that the flows considered have no swirl, i.e. the azimuthal

velocity uθ = 0, and that both red and blue fluids have equal densities for the

sake of simplicity. With the aid of the continuity equation (3), one can rewrite

Eq.(4) as [53]

ρ [∂tuα + ∂β(uαuβ)] = −∂αp+ ∂β [µ (∂βuα + ∂αuβ)]

+µ (∂ruα + ∂αur)

r− 2µur

r2δαr −

ρuαurr

. (5)

The term on the right-hand side of Eq.(3) and the last three terms on the right-

hand side of Eq.(5) arise from the cylindrical polar coordinates, and they are

hereafter referred to as the additional terms. In order to recover these terms, we

6

follow the previous works [54, 34, 35] and define the single-component collision

operator as

(Ωki )

(1) = − 1

τ

[

fki (~x, t)− fk,eq

i (~x, t)]

+ δthki (~x+ ~eiδt/2, t+ δt/2), (6)

which uses the standard Bhatnagar-Gross-Krook (BGK) approximation where

the PDF fki is relaxed towards its equilibrium distribution function fk,eq

i with a

single relaxation time τ . For the two-dimensional 9-velocity (D2Q9) model [55],

the lattice velocity ~ei ≡ (eir, eiz) is defined as ~e0 = (0, 0), ~e1,3 = (±c, 0), ~e2,4 =

(0,±c), ~e5,7 = (±c,±c), and ~e6,8 = (∓c,±c), where c = δx/δt is the lattice speed

and δx the lattice spacing. The equilibrium distribution function is obtained

by a Taylor expansion of Maxwell-Boltzmann distribution with respect to the

velocity ~u:

fk,eqi = ρkwi

[

1 +~ei · ~uc2s

+(~ei · ~u)22c4s

− ~u2

2c2s

]

, (7)

where ρk is the local density of the fluid k, cs = c/√3 is the speed of sound, and

the weight coefficients wi are given by w0 = 4/9, w1−4 = 1/9 and w5−8 = 1/36.

Conservation of mass for each fluid and total momentum conservation require

ρk =∑

i

fki =

∑

i

fk,eqi , (8)

ρuα =∑

k

∑

i

fki eiα =

∑

k

∑

i

fk,eqi eiα, (9)

where ρ =∑

k ρk.

The source term hki in Eq.(6) is introduced to recover the additional terms

in the continuity and momentum equations [i.e., Eqs.(3) and (5)], and it is given

by

hki = −wiρkurr

+1

c2swieiαH

kα, (10)

with

Hkα =

νρk (∂ruα + ∂αur)

r− 2νρkur

r2δαr −

ρkuαurr

, (11)

where ν = µ/ρ is the kinematic viscosity of the fluid mixture.

Using the Chapman-Enskog multiscale expansion, the continuity and mo-

mentum equations can be exactly derived from Eqs.(1), (2) and (6)-(11) in

7

absence of the perturbation and recoloring operators, where the pressure and

the fluid viscosity are given by (see Appendix A for the derivation)

p = ρc2s, (12)

µ = ρc2s

(

τ − 1

2

)

δt. (13)

According to Eq.(6), all of the terms in hki , given by Eqs.(10) and (11), take

their values at the position ~x + ~eiδt/2 and time t + δt/2, which is known as

centered scheme. In order to avoid the implicitness of the evolution equations,

the source term is simply evaluated by

hki (~x+ ~eiδt/2, t+ δt/2) = hki (~x, t) (14)

in the practical simulations, as previously done in Refs. [56, 57, 34, 35]. It

is shown in Appendix B that such a simple treatment allows us to accurately

account for axisymmetric contributions in the continuity and momentum equa-

tions, consistent with the previous findings in single-phase and multiphase flow

simulations [58, 56, 57, 34, 35].

To allow for unequal viscosities of the two fluids, we determine the viscosity

of the fluid mixture by a harmonic mean [59, 41]

1

µ (ρN )=

1 + ρN

2µR+

1− ρN

2µB, (15)

where µk (k = R or B) is the dynamic viscosity of fluid k; and ρN is the

color function (or phase-field function), which is used to describe the spatial

distribution of the two fluids and is defined as

ρN (~x, t) =ρR(~x, t)− ρB(~x, t)

ρR(~x, t) + ρB(~x, t), −1 ≤ ρN ≤ 1. (16)

It has been shown that the choice of Eq.(15) can ensure a constant viscosity

stress across the interface, resulting into a higher accuracy than other choic-

es [41].

In the perturbation step, the continuum surface force (CSF) model is used

to model the interfacial tension, which has been demonstrated to greatly reduce

spurious velocities and improve the isotropy of the interface [10, 51, 60]. In the

8

CSF model, a volume-distributed interfacial force ~fs is added in the momentum

equation to induce the local stress jump across the interface. The interfacial

force acts centripetally normal to the local interface with a magnitude propor-

tional to the gradient of the color function, and its expression in 3D is

~fs(~x, t) = −1

2σ∇ ·

(

~n

|~n|

)

∇ρN , (17)

where σ is an interfacial tension parameter, and ~n is the interface normal vector

defined by ~n = ∇ρN .

In the axisymmetric case, there is an extra term in the interfacial force ~fs,

that is

~fs(~x, t) = −1

2σ∇c ·

( ∇cρN

|∇cρN |

)

∇cρN − 1

2σ

∂rρN

r|∇cρN |∇cρN , (18)

where ∇c is the gradient in the cylindrical coordinates given by ∇c = (∂r, ∂z).

It is noted in the above equation that the first term on the right-hand side is

the one adopted by the color-gradient model in two dimensions, and that the

last term is the extra term responsible for the three dimensionality. Following

the previous works [51, 45], the 2D curvature ∇c ·(

∇cρN

|∇cρN |

)

in Eq.(18) can be

rewritten as

∇c ·( ∇cρ

N

|∇cρN |

)

= nrnz (∂rnz + ∂znr)− n2r∂znz − n2

z∂rnr, (19)

where nα = ∂αρN√(∂rρN )2+(∂zρN )2

.

The interfacial force Eq.(18) can be incorporated into the LBM through

different schemes. In the present study, the force model of Guo et al. [61] is

employed for its high accuracy in modeling a spatially varying body force and

capability in reducing effectively spurious velocities. According to Guo et al.,

the perturbation operator responsible for generating the interfacial tension is

expressed by [45]

(Ωki )

(2) = Ak

(

1− 1

2τ

)

wi

(

eiα − uαc2s

+eiβuβc4s

eiα

)

fsα(~x, t)δt, (20)

where Ak is the fraction of interfacial tension contributed by the fluid k, and

satisfies∑

k Ak = 1. In the presence of the interfacial tension force, the velocity

9

should be re-defined to correctly recover the Navier-Stokes equations, i.e.,

ρ(~x, t)uα(~x, t) =∑

k

∑

i

fki (~x, t)eiα +

1

2fsα(~x, t)δt. (21)

Although the perturbation operator generates an interfacial tension, it does

not ensure the immiscibility of both fluids. To promote phase segregation and

maintain the interface, the recoloring algorithm proposed by Latva-Kokko and

Rothman [47] is applied. This algorithm allows the red and blue fluids to mix

moderately at the tangent of the interface, and at the same time keeps the color

distribution symmetric with respect to the color gradient. Thus, it can further

reduce spurious velocities and remove the lattice pinning problem arising in

the original recoloring operator of Gunstensen et al. [9]. By replacing the 3D

gradient ∇ with its axisymmetric counterpart ∇c, one can obtain the recoloring

operator in the axisymmetric case, which reads as

(ΩRi )

(3)(

fR‡i

)

=ρRρf †i (~x, t) + β

ρRρBρ

wi~ei · ∇cρ

N

|~ei||∇cρN | ,

(ΩBi )

(3)(

fB‡i

)

=ρBρf †i (~x, t)− β

ρRρBρ

wi~ei · ∇cρ

N

|~ei||∇cρN | ,(22)

where f †i is the post-perturbation value of the total distribution function; fR‡

i

and fB‡i are the post-segregation (recolored) distribution functions of the red

and blue fluids, respectively; β is a free parameter associated with the interface

thickness and takes a value between zero and unity. In this study, β is taken

as 0.7 to maintain a steady interface [52], which corresponds to an interface

thickness of 4 to 5 lattices. In addition, a previous study also showed that this

choice is necessary to reproduce correct droplet behavior [12].

In contrast to the BGK approximation, the multiple-relaxation-time (MRT)

model is able to enhance numerical stability and reduce spurious velocities at

the phase interface by tuning the adjustable relaxation parameters. Thus, it has

been widely used in various multiphase/multicomponent LBMs instead of the

BGK approximation [62, 63, 48, 49, 37, 38]. With the MRT model, the single-

component collision operator and the perturbation operator can be written as

(Ωki )

(1) = −∑

j

(

M−1S)

ij

[

mkj (~x, t)−mk,eq

j (~x, t)]

+ δthki (~x, t), (23)

10

(Ωki )

(2) = −∑

j

[

M−1

(

I− 1

2S

)]

ij

F kj (~x, t), (24)

where the transformation matrix M is given by [64]

M =

1 1 1 1 1 1 1 1 1

−4 −1 −1 −1 −1 2 2 2 2

4 −2 −2 −2 −2 1 1 1 1

0 1 0 −1 0 1 −1 −1 1

0 −2 0 2 0 1 −1 −1 1

0 0 1 0 −1 1 1 −1 −1

0 0 −2 0 2 1 1 −1 −1

0 1 −1 1 −1 0 0 0 0

0 0 0 0 0 1 −1 1 −1

, (25)

Through the transformation matrix M, the PDF fki and its equilibrium distri-

bution fk,eqi can be projected onto the moment space as mk

i =∑

j Mijfkj and

mk,eqi =

∑

j Mijfk,eqj . The resulting equilibrium distribution function in the

moment space is given by

mk,eq =(

mk,eq0 ,mk,eq

1 ,mk,eq2 , · · · ,mk,eq

8

)T

= ρk(

1,−2 + 3(u2r + u2z), 1− 3(u2r + u2z), ur,−ur, uz,−uz, u2r − u2z, uruz)T.(26)

In Eqs.(23) and (24), I is a 9×9 identity matrix and S is a non-negative diagonal

matrix defined by

S = diag (s0, s1, s2, s3, s4, s5, s6, s7, s8) , (27)

where the element si represent the inverse of the relaxation time for the trans-

formed PDF mki as it is relaxed to the equilibrium distribution function in the

moment space,mk,eqi . The parameters s0, s3 and s5 correspond to the conserved

moments (i.e., density and momentum) and have no effect on the derivation of

the NSEs [65]. For simplicity, we choose s0 = s3 = s5 = 0. s1 determines the

bulk viscosity ζ through

ζ =

(

1

s1− 1

2

)

c2sδt, (28)

11

and it is considered as an adjustable parameter since the binary fluids are in-

compressible. s7 and s8 are related to the viscosity of fluid mixture by

s7 = s8 =1

τ, and µ = ρc2s

(

τ − 1

2

)

δt. (29)

Besides, symmetry requires that s4 = s6. Consequently, three independent

parameters s1, s2 and s4(= s6) can be freely adjusted to enhance the stability

of MRT model [66, 50, 62, 63]. Following the guidelines and suggestions in

Ref. [64], we choose s1 = 1.63, s2 = 1.14, and s4 = 1.92 in this study. It

was also demonstrated that such a choice can effectively suppress unphysical

spurious velocities in the vicinity of the contact line, resulting in an increased

numerical accuracy in simulating contact angles [50]. The MRT forcing term

Fk is given by [63]

Fk =(

F k0 , F

k1 , F

k2 , · · · , F k

8

)T= Ak (0, 6(urfsr + uzfsz),−6(urfsr + uzfsz),

fsr,−fsr, fsz,−fsz, 2(urfsr − uzfsz), urfsz + uzfsr)T . (30)

Finally, we note in the MRT framework that, the recoloring operator is kept the

same as the one in the BGK framework, as previously given by Eq.(22), because

it does not contain any terms related to the relaxation time.

3. Numerical Validations

In this section, the axisymmetric color-gradient LBM proposed in Sect.2

is validated by three typical cases, namely static droplet test, oscillation of a

viscous droplet, and breakup of a liquid thread. In each of the simulations below,

r = 0 represents the axis of symmetry, and the singularity will occur at r = 0

because of the terms containing r−1 [54, 36]. To avoid the singularity, we set

the first lattice line at r = 0.5δx and apply the symmetry boundary condition

to a ghost lattice line positioned at r = −0.5δx (see Fig.1):

fR‡1 (P ) = fR‡

3 (Q), fR‡5 (P ) = fR‡

6 (Q), fR‡8 (P ) = fR‡

7 (Q),

fB‡1 (P ) = fB‡

3 (Q), fB‡5 (P ) = fB‡

6 (Q), fB‡8 (P ) = fB‡

7 (Q),(31)

12

Figure 1: (Color Online) Schematic diagram of the computational geometry and setup of the

boundary conditions.

where Q is an arbitrary node at the first fluid line, and P is the symmetric ghost

node of Q. For the solid wall, no-slip boundary condition is enforced using the

halfway bounce-back scheme [67], which means the particles that hit the solid

wall, then simply return back in the opposite direction where they came from.

Specifically, as shown in Fig.1, the unknown PDFs at the fluid node ~xf adjacent

to the solid wall are determined by

fR3 (~xf , t+ δt) = f

R‡1

(~xf , t), fR6 (~xf , t+ δt) = f

R‡8

(~xf , t), fR7 (~xf , t+ δt) = f

R‡5

(~xf , t),

fB3 (~xf , t+ δt) = f

B‡1

(~xf , t), fB6 (~xf , t+ δt) = f

B‡8

(~xf , t), fB7 (~xf , t+ δt) = f

B‡5

(~xf , t).

(32)

The partial derivatives in the source term hki and the interfacial force ~fs should

be evaluated via suitable difference schemes. To minimize the discretization

errors, the fourth-order isotropic finite-difference scheme,

∂αψ(~x) =1

c2sδt

∑

i

wiψ(~x+ ~eiδt)eiα, (33)

13

is used to evaluate the derivatives of a variable ψ at ~x 6= ~xf ; whereas at the

fluid node ~xf we impose the derivative terms to be zero in the evaluation of the

interfacial force, and use the second-order difference schemes to evaluate the

derivative terms in hki , i.e.

∂rψ(~xf ) = − 1

3δx[3ψ(~xf ) + ψ(~xf + ~e3δt)] ,

∂zψ(~xf ) =1

2δx[ψ(~xf + ~e2δt)− ψ(~xf + ~e4δt)] , (34)

which is obtained on the basis of the zero velocity condition at the solid wall.

In addition, it is worth mentioning that, in what follows, all of the simulation

results are obtained by our proposed LBM in Sect.2 unless otherwise noted.

3.1. Static droplet test

The static droplet test represents a traditional benchmark of two-phase flow

models. It consists of a ‘spherical’ droplet (red fluid) initially located at the

centre of the lattice domain with 100× 200 lattices in the rz-plane. The bound-

ary conditions for both fRi and fB

i are periodic in the z-direction and the right

boundary is the solid wall where the no-slip boundary condition is imposed.

According to the Laplace’s law, when the system reaches the equilibrium state,

the pressure difference between the interior and exterior of the droplet ∆p is

related to the interfacial tension σ by

∆p =2σ

RD, (35)

where RD is the radius of the droplet. Fig.2 shows the pressure difference

∆p against R−1D using the following parameters: ρR = ρB = 1, µR = 0.3,

µB = 0.03, and σ = 0.01. It can be found that our LBM results (represented by

hollow circles) are in excellent agreement with the Laplace’s law (represented

by the solid line). Based on the recoloring operator Eq.(22), one can derive an

analytical expression for the equilibrium interface profile:

ρN (r, z) = tanh

(

RD −√

(r − rc)2 + (z − zc)2

ξ

)

, (36)

14

1/RD

∆p

0 0.02 0.04 0.06 0.08 0.1 0.120

0.0005

0.001

0.0015

0.002

Figure 2: (Color Online) Comparison of the LBM results (represented by discrete symbols)

with the Laplace’s law (represented by the solid line) for pressure jump across a static droplet

interface. Note that the red circles represent the simulation results from the color-gradient

LBM proposed in Sect.2, while the green triangles represent the simulation results from the

Li-based model that is presented in Appendix B.

15

r

ρN

0 20 40 60 80 100

-1

-0.5

0

0.5

1 λ=1λ=10λ=102

λ=103

λ=104

Theoretical

Figure 3: (Color Online) The profiles of the color function at different values of λ for a static

droplet with RD = 40. The discrete symbols represent the simulation results of the present

LBM and the solid line is the theoretical profile given by Eq.(36).

16

where rc and zc are the coordinates of the centre of the droplet, and ξ is a

measure of the interface thickness related to the segregation parameter β by

ξ = 1/(6kβ) [68]. Here k is a geometric constant that is determined by

2k =∑

i

wieiαeiβ|~ei|

. (37)

From Eq.(37), one can easily obtain k = 19+

118

√2≈ 0.1504 for the D2Q9 lattice.

Next, we conduct a series of LBM simulations for RD = 40 over a broad range

of viscosity ratios (λ = µR

µB), and compare the simulated equilibrium interface

profiles with the analytical one given by Eq.(36). In these simulations, all the

parameters are kept the same as those used in Fig.2 except µB, which is varied

to obtain different viscosity ratios. Fig.3 displays the color function ρN as a

function of the distance from the droplet centre for λ = 1, 10, 102, 103, 104. Itcan be clearly seen that the LBM simulations are stable for the viscosity ratios

up to 104, which is much higher than the highest viscosity ratio that other

multicomponent LBMs can achieve, e.g., the highest values of the viscosity

ratio are only on the order of 6 and tens for the interparticle-potential model

and the free-energy model, respectively [27]. Also, the equilibrium interface

profiles are all in good agreement with the analytical solution, indicating that

our axisymmetric multicomponent LBM can correctly model and capture phase

interface. It can be seen from the analytical expression of interface profile that

the interface thickness is only determined by the segregation parameter β and is

independent of the choice of BGK or MRT collision models (provided that both

models produce stable numerical results). This is confirmed by our numerical

results presented in Fig.4, where (a) compares the MRT results of λ = 10

and 104 with the analytical predictions from Eq.(36) for β = 1 (which leads

to a thinner interface than β = 0.7), and (b) compares the numerical results

obtained by the BGK model with those by the MRT model for β = 0.7 and

λ = 10. The present color-gradient LBM is a diffuse-interface model, which

requires the Cahn number Cn = ξ/RD ≪ 1 to recover a sharp-interface limit.

Since ξ = 16kβ = 1

6×0.1504×0.7 = 1.583 is a constant when β is fixed at 0.7 (for

a larger β, e.g. β = 1, the interface cannot be accurately described because of

17

r

ρN

0 20 40 60 80 100

-1

-0.5

0

0.5

1 λ=10λ=104

Theoretical

(a)

r

ρN

0 20 40 60 80 100

-1

-0.5

0

0.5

1 BGKMRTTheoretical

(b)

Figure 4: (Color Online) The profiles of the color function obtained by (a) the MRT model

at β = 1 and λ = 10, 104 and (b) the BGK and MRT models at β = 0.7 and λ = 10, for a

static droplet with RD = 40. The discrete symbols represent the simulation results and the

solid lines are the theoretical profiles given by Eq.(36).

insufficient number of lattice grids across the interface, which can be seen in

Fig.4(a) and Ref. [52]), small value of Cn can be only achieved by increasing

the grid number of droplet size or computational domain, which will largely

increase the computing cost. To strike a balance between the computing cost and

accuracy, we take RD = 40 lattices so that Cn = 0.0396, which was previously

demonstrated small enough to provide satisfactory predictions of the droplet

dynamics in color-gradient LBMs [41, 42, 38, 69].

A numerical artifact observed in many numerical methods is the presence

of spurious velocities at the phase interface. Spurious velocities are small-

amplitude artificial velocity fields arising from an imbalance between discretized

forces in the interfacial region. Spurious velocities on one hand may prevent the

system from reaching a true equilibrium state, and on the other hand, they can

sometimes be as large as the characteristic velocity of the flow problem, leading

to numerical instability and/or contamination of physical velocities. Several

attempts have been made to identify the cause of spurious velocities and to

reduce their magnitude, see Refs. [51, 10] and a general review by Connington

and Lee [70]. However, none of the existing color-gradient LBMs can elimi-

18

nate the spurious velocities to roundoff. This is also true for the present LBM.

Table 1 shows the maximum spurious velocities (|~u|max) at various viscosity

ratios, where the values of |~u|max are magnified by 105 times. It is seen that

the maximum spurious velocities increase with the viscosity ratio, and that all

of them are on the order of 10−4 or even smaller, comparable to those produced

by the original 2D color-gradient model (see the bottom row in Table 1). Also,

the present spurious velocities are much smaller than those obtained with the

commonly-used interparticle-potential model [71, 40].

Table 1: The maximum spurious velocities (|~u|max) at various viscosity ratios for RD = 40

and µR = 0.3.

λ 1 10 102 103 104

|~u|max × 105Axisymmetic LBM 0.223 0.229 1.327 8.533 26.548

Original 2D LBM 0.204 0.206 0.752 5.435 19.952

3.2. Oscillation of a viscous droplet

Droplet oscillation is often used to assess whether an axisymmetric multi-

phase/multicomponent model is able to simulate dynamic problems. A droplet

(red fluid) that is slightly deformed to be an axisymmetric ellipsoid, is immersed

in a second viscous fluid (blue fluid). Upon release, the ellipsoidal droplet starts

to oscillate due to the imbalanced interfacial tension forces. For the droplet os-

cillation, Miller and Scriven [72] derived an analytical solution for the oscillation

frequency of the nth mode,

ωn = ω∗n − 1

2a√

ω∗n +

1

4a2, (38)

where ω∗n is Lamb’s natural resonance frequency,

ω∗n =

√

n(n+ 1)(n− 1)(n+ 2)

R3D [nρB + (n+ 1)ρR]

σ, (39)

and RD is the radius of the droplet at equilibrium. In Eq.(38), the last two

terms represent the corrections to ω∗n due to viscous effects, and the parameter

19

-100 -50 0 50 1000

50

100

150

200

250

300

t=13000-100 -50 0 50 100

0

50

100

150

200

250

300

t=105

-100 -50 0 50 1000

50

100

150

200

250

300

t=0-100 -50 0 50 100

0

50

100

150

200

250

300

t=6500

Figure 5: (Color Online) Evolution of the shape of an oscillating droplet for Rr = 40, Rz = 55,

µR = 2× 10−2, µB = 2× 10−3, and σ = 5× 10−3.

20

t

Rr/R

D

0 20000 40000 60000 80000

0.9

1

1.1σ=2×10-2

σ=10-2

σ=5×10-3

σ=10-3

Figure 6: (Color Online) Time evolution of the half-axis length Rr at different values of σ for

Rr = 40, Rz = 55, µR = 2 × 10−2, and µB = 2× 10−3. Note that the half-axis length Rr is

normalized by the equilibrium radius RD.

21

a is given by

a =(2n+ 1)2

√ρRρBµRµB√

2[nρR + (n+ 1)ρB](√ρRµR +

√ρBµB)

. (40)

As Refs. [31, 34, 36], the second mode of the oscillation is considered in this work,

i.e., n = 2. The simulations are conducted in a 150 × 300 lattice domain with

the boundary conditions the same as in the static droplet test. The densities

of red and blue fluids are both fixed at ρc = 1. Following Liang et al. [36], the

interface profile is initialized as

ρN (r, z) = tanh

(

RD1−

√

(r − rc)2/R2r + (z − zc)2/R2

z

ξ

)

, (41)

where (rc, zc) = (0, 150) is the center position of the ellipsoidal droplet; Rr and

Rz are the half-axis lengths of the ellipsoid in the r and z directions, respectively;

and the equilibrium radius RD is calculated by RD = (R2rRz)

1/3 (based on mass

conservation of the droplet). With the initial interface profile, i.e. Eq.(41),

one can determine the initial density fields by ρR = ρc(1 + ρN )/2 and ρB =

ρc(1 − ρN )/2. Fig.5 shows the snapshots of an oscillating droplet at different

times for Rr = 40, Rz = 55, µR = 2× 10−2, µB = 2× 10−3, and σ = 5× 10−3.

It is observed that the droplet oscillates with time until finally reaching an

equilibrium spherical shape. The droplet oscillation is quantified by measuring

the half-axis length Rr as a function of time, which is plotted in Fig.6 for four

different values of σ, i.e., σ = 2 × 10−2, 10−2, 5 × 10−3, and 10−3. Note in

these cases that all the parameters are kept the same as those used in Fig.5

except σ. We can clearly see that Rr fluctuates around RD during the droplet

oscillation for all values of σ, but its maximum value decreases with decreasing σ.

Besides, the oscillation period decreases as the interfacial tension is increased.

Specifically, the oscillation periods TLBM computed by the present LBM are

6449, 9114.3, 13063.2 and 30046.8, respectively, for σ = 2×10−2, 10−2, 5×10−3

and 10−3, and their corresponding analytical solutions Tanal are 6266.6, 8929.1,

12739.7, and 29260, where Tanal = 2π/ω2 and ω2 is given by Eq.(38). Evidently,

the numerical predictions TLBM agree well with the analytical solutions Tanal,

with a maximum relative error of around 2.9%.

22

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

++++++++++++++++

++++++++++++

+++++++++

++++++++ +

++++

+ ++++

+++

++

++

++

++

++ +

+ + + + + + + + + +

t

Rr/R

D

0 20000 40000 60000 80000

0.9

1

1.1

Rr=30, Rz=45Rr=40, Rz=55Rr=30, Rz=45 (Li-based model)Rr=40, Rz=55 (Li-based model)

+

Figure 7: (Color Online) Time evolution of the half-axis length Rr (normalized by the equi-

librium radius RD) at different initial sizes of the droplet for µR = 2× 10−2, µB = 2× 10−3,

and σ = 5 × 10−3. Note that the discrete symbols represent the simulation results from the

color-gradient LBM proposed in Sect.2, while the lines represent the simulation results from

the Li-based model presented in Appendix B.

23

t

Rr/R

D

0 20000 40000 60000 80000

0.9

1

1.1λ=10λ=102

λ=103

λ=104

Figure 8: (Color Online) Time evolution of the half-axis length Rr at different values of λ

for Rr = 40, Rz = 55, µR = 0.05, and σ = 5 × 10−3. Note that the half-axis length Rr is

normalized by the equilibrium radius RD.

Next, we examine the influence of the droplet size on the oscillation period.

Fig.7 shows the temporal evolution of the half-axis length Rr at µR = 2× 10−2,

µB = 2 × 10−3, and σ = 5 × 10−3 for two different initial sizes of the droplet:

Rr = 30, Rz = 45 and Rr = 40, Rz = 55. It is seen that decreasing the

droplet size decreases the oscillation period. The computed oscillation periods

for the large and small droplets are respectively 13063.2 and 9003.2, which are

very close to their corresponding analytical values 12739.7 and 8674.5, with the

relative errors within 3.8%.

Then, we further examine the method’s capability for simulating the binary

fluids with high viscosity ratio. Four different viscosity ratios are considered:

λ = 10, 102, 103 and 104, and they are achieved by adjusting µB while keeping

24

r

z

0 20 40 600

10

20

30

40

50

60

70

Figure 9: (Color Online) Velocity field at t = 20000 timesteps for Rr = 40, Rz = 55,

µR = 0.05, λ = 104, and σ = 5× 10−3. Note that the velocity vectors are obtained by using

the MRT model of Pooley et al. [48], and only a part of computational domain is illustrated

in order to clearly show the local abnormal velocities. The red lines represent the droplet

interface, while the pink line is the axis of symmetry (r = 0).

25

µR fixed at 0.05. Other parameters are given as follows: Rr = 40, Rz = 55, and

σ = 5 × 10−3. Fig.8 illustrates the temporal evolution of the half-axis length

Rr at various viscosity ratios. We can see that the simulations are stable for all

the viscosity ratios considered, and the amplitude of oscillation (represented by

Rr) increases with λ since at larger λ (which corresponds to smaller µB), the

viscous damping effect is reduced. In addition, as the viscosity ratio is increased

the oscillation period decreases, but the decrease is insignificant for λ ≥ 102.

This can be more clearly seen in Table 2, which shows the comparison between

TLBM and Tanal at various viscosity ratios. Overall, the computed oscillation

periods are in agreement with the analytical results for all viscosity ratios, with

a maximum relative error of around 6.2%. Also, the relative error increases

with increasing viscosity ratio, indicating that the present axisymmetric LBM

has a lower predictive accuracy (although still acceptable) at higher viscosity

ratio. It is interesting to make a remark concerning the choice of free parame-

ters s1, s2 and s4 in the diagonal relaxation matrix S. These free parameters

are not uniquely determined although Lallemand and Luo [64] provided some

guidelines to choose some of them. It is therefore not surprising that a number

of forms of diagonal relaxation matrix have been reported in literature with

different free parameters, e.g., Refs. [73, 74, 63, 75, 62, 48, 76]. To date, a trial-

and-error approach is still required to find the most stable and reliable set of

free parameters for a specific problem. For the simulations of droplet oscilla-

tion, we have found that the present MRT model with the choice of s1 = 1.63,

s2 = 1.14, and s4 = 1.92 produces more stable results than the well-known

Two-Relaxation-Time (TRT) algorithm [73] (s1 = s2 = 1τ , s4 = 8(2−s1)

8−s1), the

MRT of Pooley et al. [48] (s1 = s2 = s4 = 1), and the MRT of Fakhari and

Lee [76] (s1 = s2 = 1, s4 = 1.7). For example, in the case of λ = 104, the

simulation is stable at all times for the present MRT model, but diverges at

approximately 19000 timesteps for the TRT, 230000 timesteps for the MRT of

Pooley et al. and 92000 timesteps for the MRT of Fakhari and Lee, respec-

tively. Note that, although the simulation does not blow up until t = 230000

when the MRT of Pooley et al. is used, some abnormal velocities have been

26

clearly observed near the axis of symmetry and the droplet interface inside the

blue (less viscous) fluid at an earlier time i.e. t = 20000 (see Fig.9). The mag-

nitude of abnormal velocities increases gradually with time, eventually leading

to divergence of the simulation. In addition, the MRT model should be more

stable than the BGK model even if the choice of free parameters is not opti-

mal [64, 73, 62, 63, 50]. This is demonstrated by our simulations, in which the

BGK model breaks down at t = 4000, much earlier than all of the MRT models

we have tested (note that the model will be more unstable if it breaks down ear-

lier). Based on the limited numerical comparisons shown here, the guidelines

given by Lallemand and Luo [64] (recall that our free parameters are chosen

following their recommendation) seem to be applicable as well in the present

color-gradient MRT model. However, further study is required to provide more

insights in determining optimal parameters.

Table 2: Comparison between the computed (TLBM ) and analytical (Tanal) oscillation periods

at various viscosity ratios for Rr = 40, Rz = 55, σ = 5× 10−3, and µR = 0.05.

λ 10 102 103 104

TLBM 13448.1 13008.9 12890.0 12824.0

Tanal 13144.8 12456.2 12174.8 12077.7

Er = |TLBM−Tanal||Tanal| × 100% 2.31% 4.44% 5.87% 6.19%

In addition to the MRT model, which can enhance the stability for simu-

lation of multiphase/multicomponent flows with high viscosity ratio, we also

implement a variant of the BGK model, which is based on a regularization of

the pre-collision PDFs. This model is known as the RLB model, which was

proposed independently by Latt and Chopard [77], Chen et al. [78], and Zhang

et al. [79]. It was found that, in comparison with the BGK model, the RLB

model offers an improvement in both stability and accuracy without adding any

substantial complication. Later, the RLB model was also demonstrated to per-

form very well in the multicomponent interparticle-potential LBM [80]. In view

of the advantages of RLB model, it is worthwhile to extend it to the present

27

color-gradient LBM and test its effectiveness in the simulation of axisymmetric

multicomponent flows with high viscosity ratio. Inspired by the previous work-

s [77, 81, 82], we derive a regularized expression for computing the incoming

PDFs before collision, which is given by

fki ≈ fk,eq

i + fk,(1)i = fk,eq

i +wi

2c4sQiαβΠ

k,neqαβ − wi

2c2seiαf

ksα, (42)

where Qiαβ = eiαeiβ − c2sδαβ , fksα = Akfsα, and Πk,neq

αβ =∑

i(fki − fk,eq

i )eiαeiβ

is the nonequilibrium component of the momentum flux tensor. It can be easily

proved that fk,(1)i in Eq.(42) satisfies the following constraints:

∑

i

fk,(1)i = 0,

∑

i

fk,(1)i eiα = −1

2fksα,

∑

i

fk,(1)i eiαeiβ = −ρkc2sτ (∂βuα + ∂αuβ)−

1

2

(

uαfksβ + uβf

ksα

)

.

(43)

Except for an additional, straightforward regularization step as shown in Eq.(42),

all the constituents of the collision operator are kept the same as those in the

color-gradient BGK model, i.e. Eqs.(6), (20) and (22). We finally test the color-

gradient RLB model’s capability through the simulation of droplet oscillation

with λ = 104. All the parameters and boundary conditions are identical to

those used in Fig.9. It is found that the RLB model produces stable results at

all times, and the obtained results are in excellent agreement with the results

of our MRT model that uses the free parameters recommended by Lallemand

and Luo [64] (see Fig.10). This suggests that the color-gradient RLB model is

effective for the quantitative study of axisymmetric multicomponent flows, and

is numerically accurate and stable even for very small relaxation time τ and

high viscosity ratio, which remain a challenge for many of the MRT models.

3.3. Breakup of a liquid thread

In order to reveal the capability of the present LBM in the simulation of

large topological changes, we consider the problem of the breakup of a liquid

28

t

Rr/R

D

0 20000 40000 60000 80000 100000 120000 140000

0.9

1

1.1

Figure 10: (Color Online) Time evolution of the half-axis length Rr (normalized by the

equilibrium radius RD) for Rr = 40, Rz = 55, µR = 0.05, λ = 104, and σ = 5 × 10−3.

Note that the discrete symbols represent the simulation results from the color-gradient RLB

model, while the lines represent the simulation results from the color-gradient LBM proposed

in Sect.2.

29

thread into multiple droplets. This problem was first studied experimentally

by Plateau [83] and later investigated theoretically by Rayleigh [84]. Through

a linear stability analysis, Rayleigh [84] showed that a cylindrical liquid thread

of radius Rc is unstable if the wavelength λd of a disturbance is greater than

its circumference 2πRc. In other words, the liquid thread is unstable when the

wave number k∗ (k∗ = 2πRc/λd) is less than unity, and stable when k∗ is larger

than unity. Several preliminary numerical tests conducted with k∗ > 1 do show

that the liquid thread does not break up, consistent with the linear stability

theory. In the following simulations, we will focus on the cases with breakup

(i.e., k∗ < 1), and compare the simulated results with some available literature

data.

The simulations are run in a 300× λd lattice domain with the same bound-

ary conditions as in the static and oscillating droplet tests. A disturbance is

introduced by specifying the initial interface profile as

ρN (r, z) = tanhRc + d− r

ξ, (44)

where d is the interfacial disturbance imposed on the liquid thread and is given

by d = 0.1Rc cos (2πz/λd). In the absence of disturbance, Eq.(44) represents a

perfect cylindrical column with radius Rc, which is taken as 60 in our study.

Note that the disturbance can be also introduced in other forms, e.g., a fluctu-

ation on the initial velocity field in Ref. [35]. Seven cases with the wavelengths

λ = 420, 500, 600, 800, 1000, 1200, 1800 are considered, which correspond to

seven different wave numbers, i.e. k∗ = 0.90, 0.75, 0.63, 0.47, 0.38, 0.31, 0.21.The densities of both fluids are set to be unity, and the interfacial tension σ is

6 × 10−3. The Ohnesorge number, which is defined by Oh = µR/√ρRσRc, is

kept at 0.1. Using the aforementioned parameters, one can easily get µR = 0.06,

and µB is then determined by the viscosity ratio λ, which is chosen as 10 unless

otherwise stated.

Fig.11 shows the snapshots of the breakup of a liquid thread for three typical

wave numbers, where time is normalized by the capillary time, tcap =√

R3cρR/σ.

It is observed that the interfacial disturbances gradually grow with time for all

30

Figure 11: (Color Online) Snapshots of the breakup of a liquid thread at λ = 10 for different

wave numbers, left panel: k∗ = 0.63, middle panel: k∗ = 0.31, right panel: k∗ = 0.21. Time

is normalized by the capillary time√

R3cρR/σ. Note that the cyan solid lines represent the

simulation results from the color-gradient LBM proposed in Sect.2, while the red dashed lines

represent the simulation results from the Li-based model presented in Appendix B.

31

k*

R*

0 0.2 0.4 0.6 0.8 10

1

2

3

4 LBM, MainLBM, SatelliteAshgriz, Num., MainAshgriz, Num., SatelliteLafrance, Anal., MainLafrance, Anal., SatelliteLafrance, Exp., MainLafrance, Exp., SatelliteRutland, Exp., MainRutland, Exp., Satellite

Figure 12: (Color Online) R∗ as a function of the wave number for λ = 10. R∗ is the droplet

radius normalized by Rc.

32

of the wave numbers. Then the liquid thread in the middle region progressively

thins and, at the same time the ends enlarge until the thread breaks up, forming

a thin ligament as well as a main droplet (see, for example, time 25.83 at k∗ =

0.21). Afterwards, the ligament contracts continuously due to the dominant

interfacial tension, eventually leading to a spherical satellite droplet formed.

During the contraction of the liquid ligament, a pair of droplets can be found at

the end of the ligament, and they exhibit an increasing trend to pinch-off from

the middle portion as the wave number increases. Therefore, it is expected that

a liquid ligament can break up into multiple droplets as long as it is sufficiently

long, consistent with the previous findings [85, 86, 36]. In addition, we also

quantify the sizes of the main and satellite droplets at various wave numbers,

which are plotted in Fig.12. For the purpose of comparison, previous results

including the finite element results by Ashgriz and Mashayek [85], the analytical

solutions and experimental data of Lafrance [87], as well as the experimental

data of Rutland and Jameson [88] are also presented. Evidently, the main

and satellite droplets both decrease in size with increasing the wave number;

and also, our simulation results show good agreement with available literature

data in general, providing further validation of the present axisymmetric color-

gradient LBM.

To show the influence of the viscosity ratio on the breakup of the liquid

thread, we also simulate the case of λ = 102, which is achieved by varying µB

while keeping all the other parameters the same as used in the case of λ = 10.

Fig.13 depicts the snapshots of the breakup of a liquid thread at λ = 102 for

k∗ = 0.63 (left), 0.31 (middle), and 0.21 (right), where the snapshots are taken

at the times exactly the same as in the case of λ = 10. By comparing the cases

of λ = 10 and λ = 102, we can find that the viscosity ratio can affect not only

the interface structure but also its dynamical evolution. For example, the liquid

thread breaks up earlier at λ = 102 for each of the wave numbers, suggesting

that increasing viscosity ratio can speed up the growth of Rayleigh instability.

However, it is found that the viscosity ratio has a negligible effect on the sizes

of the main and satellite droplets formed. This could explain why the data

33

Figure 13: (Color Online) Snapshots of the breakup of a liquid thread at λ = 102 for different

wave numbers, left panel: k∗ = 0.63, middle panel: k∗ = 0.31, right panel: k∗ = 0.21. Time

is normalized by the capillary time√

R3cρR/σ.

34

from different literature sources are all comparable with those experimentally

obtained by Rutland and Jameson [88] in the k∗ − R∗ diagram (where R∗ is

the droplet radius normalized by Rc), although they have considered different

viscosity ratios.

4. Conclusions

A color-gradient LBM is developed for simulating axisymmetric multicom-

ponent flows for a broad range of viscosity ratios. Like the Cartesian color-

gradient models, this method uses a collision operator that is a combination of

three separate parts, namely the single-component collision operator, pertur-

bation operator, and the recoloring operator. In order to recover correctly the

continuity and momentum equations in each single-component region, a source

term is added into the single-component collision operator to account for the

axisymmetric contributions. In the perturbation step, an interfacial force of

axisymmetric form is derived using the CSF concept together with a coordi-

nate transformation, and is incorporated into the LBM through the body force

model of Guo et al. [61]. A recoloring operator proposed by Latva-Kokko and

Rothman [47] is extended to the axisymmetric case for producing the phase

segregation and guaranteeing the immiscibility of both fluids. To improve the

numerical stability for solving binary fluids with high viscosity ratio, the ax-

isymmetric color-gradient LBM is implemented in the MRT framework instead

of the standard BGK approximation. The usefulness of accuracy of the method

are assessed by several typical numerical examples, including static droplet test,

oscillation of a viscous droplet, and breakup of a liquid thread. The former two

examples show that the present LBM is able to accurately capture the interface,

produce low unphysical spurious velocities, and simulate the viscosity ratios up

to 104 with acceptable numerical accuracy. In the last example, the detailed

comparison between the computed results and the previous literature data shows

that the present LBM can provide reasonable predictions of the thread breakup

caused by the Rayleigh instability. In addition, the viscosity ratio is found to

35

significantly affect the growth of the Rayleigh instability, but it has a negligi-

ble effect on the sizes the formed droplets. In our future work, we would like

to apply the present color-gradient LBM for more sophisticated problems, e.g.

diesel spray formation and breakup, and compare our simulation results with

those reported by Falcucci et al. [89].

Appendix A. Derivation of axisymmetric Navier-Stokes equations us-

ing the Chapman-Enskog expansion

In the absence of the perturbation and recoloring operators, the collision

operator in Eq. (2) can be simplified as Ωki = (Ωk

i )(1), and one can rewrite

Eq. (1) as

fki (~x+~eiδt, t+ δt) = fk

i (~x, t)−fki (~x, t)− fk,eq

i (~x, t)

τ+ δth

ki (~x+~eiδt/2, t+ δt/2),

(A.1)

with k = R or B and fk,eqi given by Eq. (7). Introducing the Chapman-Enskog

expansion,

fki (~x+ ~eiδt, t+ δt) =

∞∑

n=0

ǫn

n!Dn

t fki (~x, t), (A.2)

fki =

∞∑

n=0

ǫnfk,(n)i , (A.3)

hki (~x+ ~eiδt/2, t+ δt/2) =

∞∑

n=0

(ǫ/2)n

n!Dn

t hki (~x, t), (A.4)

where ǫ = δt and Dt ≡ (∂t + ei · ∇), the following equations are obtained up to

second order in the parameter ǫ:

O(ǫ0) : fk,(0)i = fk,eq

i , (A.5)

O(ǫ1) : Dtfk,(0)i = − 1

τfk,(1)i + hki , (A.6)

O(ǫ2) : Dtfk,(1)i +

1

2D2

t fk,(0)i = − 1

τfk,(2)i +

1

2Dth

ki . (A.7)

Using Eq.(A.6), Eq.(A.7) can be written as(

1− 1

2τ

)

Dtfk,(1)i = − 1

τfk,(2)i . (A.8)

36

Note that one can use the following solvability conditions for fk,(n)i (n = 1, 2, · · · ),

∑

i

fk,(n)i = 0,

∑

k

∑

i

fk,(n)i eiα = 0, (A.9)

and the conditions for fk,eqi and hki are

∑

i

fk,eqi = ρk,

∑

i

fk,eqi eiα = ρkuα,

∑

i

fk,eqi eiαeiβ = ρkuαuβ + ρkc

2sδαβ ,

∑

i

fk,eqi eiαeiβeiγ = ρkc

2s(uαδβγ + uβδαγ + uγδαβ),

∑

i

hki eiα = Hkα,

∑

i

hki eiαeiβ = −ρkurr

c2sδαβ , (A.10)

where Hkα is defined by Eq. (11).

From Eq.(A.6) + Eq.(A.8)×ǫ, one has

Dtfk,(0)i + ǫ

(

1− 1

2τ

)

Dtfk,(1)i = − 1

τ

(

fk,(1)i + f

k,(2)i

)

+ hki . (A.11)

Summation of the above equation over i and k provides

∂tρ+ ∂α(ρuα) = −ρurr. (A.12)

In the incompressible limit, the density of the fluid mixture is assumed to be

small enough, and Eq.(A.12) can lead to the continuity equation (3).

Taking∑

k

∑

i[Eq.(A.6) + Eq.(A.8)×ǫ] yields

∂t(ρuα) + ∂βΠ(0)αβ = ∂βΓ

(1)αβ +

∑

k

Hkα, (A.13)

where

Π(0)αβ =

∑

k

∑

i

fk,(0)i eiαeiβ = ρc2sδαβ + ρuαuβ, (A.14)

and

Γ(1)αβ = −ǫ

(

1− 1

2τ

)

∑

k

∑

i

fk,(1)i eiαeiβ . (A.15)

Substitution of Eq.(11) into Eq.(A.13) results in

∂t(ρuα)+∂β(ρuαuβ) = −∂α(ρc2s)+∂βΓ(1)αβ+

µ (∂ruα + ∂αur)

r− 2µur

r2δαr−

ρuαurr

.

(A.16)

37

Applying Eq.(A.6) one can rewrite Eq.(A.15) as

Γ(1)αβ = ǫ

(

τ − 12

)∑

k

∑

iDtfk,(0)i eiαeiβ − ǫ

(

τ − 12

)∑

k

∑

i hki eiαeiβ

= ǫ(

τ − 12

)

(

∂tΠ(0)αβ + ∂γ

∑

k

∑

i fk,(0)i eiαeiβeiγ

)

+ ǫ(

τ − 12

)

ρur

r c2sδαβ

= ǫ(

τ − 12

)

[

∂tΠ(0)αβ + c2s∂γ(ρuαδβγ + ρuβδαγ + ρuγδαβ) +

ρur

r c2sδαβ

]

.(A.17)

Through the dimensional analysis, one can obtain that the ratio of the first

to the second terms in the square bracket of Eq.(A.17) has the order

O(

∂tΠ(0)αβ

c2s∂γ(ρuαδβγ + ρuβδαγ + ρuγδαβ)

)

= O(Ma2), (A.18)

where Ma is the Mach number. It shows that the term ∂tΠ(0)αβ is very small

compared with c2s∂γ(ρuαδβγ + ρuβδαγ + ρuγδαβ) and can be neglected if Ma≪1 [54]; hence Eq.(A.17) becomes

Γ(1)αβ = ǫ

(

τ − 1

2

)

[

c2s∂γ(ρuαδβγ + ρuβδαγ + ρuγδαβ) +ρurrc2sδαβ

]

= ǫ

(

τ − 1

2

)

c2s [ρ(∂βuα + ∂αuβ) + uα∂βρ+ uβ∂αρ+ uγ∂γρδαβ ]

= ǫ

(

τ − 1

2

)

c2sρ(∂βuα + ∂αuβ), (A.19)

in which we have used the continuity equation and neglected the terms of

O(Ma3). Substitution of Eq.(A.19) into Eq.(A.16) leads to

∂t(ρuα) + ∂β(ρuαuβ) = −∂α(ρc2s) + ∂β

[(

τ − 1

2

)

c2sρδt(∂βuα + ∂αuβ)

]

+µ (∂ruα + ∂αur)

r− 2µur

r2δαr −

ρuαurr

, (A.20)

which reduces to the exact momentum equation (5) in the incompressible limit

if the pressure and the viscosity are given by Eq.(12) and Eq.(13), respectively.

Appendix B. An accurate version of color-gradient model for axisym-

metric multicomponent flows

To test the accuracy of the LBM proposed in Sect.2, we also present an accu-

rate version of the color-gradient model for axisymmetric multicomponent flows.

38

In this model, the single-component collision operator and the perturbation op-

erator are derived based on the axisymmetric single-phase LBM proposed by Li

et al. [53], and the recoloring operator takes the same form as the one shown in

Eq.(22). In view of its theoretical basis, this model is called as Li-based model

here. The Li-based model can recover the correct continuity and momentum

equations in the axisymmetric coordinate system to the second-order accuracy;

and also, it does not need to introduce any approximations in the implementa-

tion. Thus, its numerical results are considered ‘accurate’ and can be regarded

as benchmark solutions. Following Li et al. [53], the single-component collision

operator and the perturbation operator are given by

(Ωki )

(1) = − 1

τ ′

[

fki (~x, t)− fk,eq

i (~x, t)]

+

(

1− 1

2τ ′

)

φki (~x, t), (B.1)

(Ωki )

(2) = Ak

(

1− 1

2τ ′

)

(eiα − uα)fsα(~x, t)

ρkc2sfk,eqi (~x, t)δt, (B.2)

with

φki =

[

(eiα − uα)

c2s

(

−2νuαr2

δαr

)

− urr

]

fk,eqi δt, (B.3)

where τ ′ = (τ + 0.5)/[1 + τδt(eiα/r)].

The macroscopic variables can be calculated by

ρk(~x, t) =

∑

i fki (~x, t)

(1 + δtur(~x,t)2r )

, (B.4)

uα(~x, t) =

∑

k

∑

i fki (~x, t)eiα + 0.5fsα(~x, t)δt

∑

k

∑

i fki (~x, t) + (δt

µr2 )δαr

. (B.5)

As seen from Eqs.(B.4) and (B.5), the densities and the fluid velocity are coupled

in a nonlinear fashion. To avoid such a nonlinear coupling, we first compute ρN

by

ρN (~x, t) =ρR(~x, t)− ρB(~x, t)

ρR(~x, t) + ρB(~x, t)=

∑

i fRi (~x, t)−∑i f

Bi (~x, t)

∑

i fRi (~x, t) +

∑

i fBi (~x, t)

. (B.6)

Once the value of ρN is obtained, the interfacial force can be calculated by

Eq.(18), and then the fluid velocity can be calculated by Eq.(B.5). Finally, the

densities of red and blue fluids are obtained through Eq.(B.4).

Figs.2, 7 and 11 show the comparison between the simulation results ob-

tained by the Li-based model and those obtained by the color-gradient LBM

39

described in Sect.2. Excellent agreement is observed in all of these figures, in-

dicating that the color-gradient LBM described in Sect.2 can provide ‘accurate’

prediction of the axisymmetric multicomponent flows albeit that the centered

scheme with an explicit approximation is used for the axisymmetric contribu-

tions. In addition, the color-gradient LBM described in Sect.2 is as simple as

the original color-gradient model in form (roughly the same except several small

additives or variants), so it is preferred and is the focus of this work.

Acknowledgements

This work is financially supported by the Thousand Youth Talents Program

for Distinguished Young Scholars, the National Natural Science Foundation of

China (No. 51506168) and the China Postdoctoral Science Foundation (No.

2016M590943). This work is also supported by the UK’s Engineering and Phys-

ical Sciences Research Council (EPSRC) under grant EP/L00030X/1.

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